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An arithmetic Hilbert–Samuel theorem for pointed stable curves

Gerard Freixas i Montplet (2012)

Journal of the European Mathematical Society

Let ( 𝒪 , , F ) be an arithmetic ring of Krull dimension at most 1 , S = Spec ( 𝒪 ) and ( 𝒳 S ; σ 1 , ... , σ n ) a pointed stable curve. Write 𝒰 = 𝒳 j σ j ( S ) . For every integer k > 0 , the invertible sheaf ω 𝒳 / S k + 1 ( k σ 1 + ... + k σ n ) inherits a singular hermitian structure from the hyperbolic metric on the Riemann surface 𝒰 . In this article we define a Quillen type metric · Q on the determinant line λ k + 1 = λ ω 𝒳 / S k + 1 ( k ...

Are rational curves determined by tangent vectors?

Stefan Kebekus, Sándor J. Kovács (2004)

Annales de l’institut Fourier

Let X be a projective variety which is covered by rational curves, for instance a Fano manifold over the complex numbers. In this paper, we give sufficient conditions which guarantee that every tangent vector at a general point of X is contained in at most one rational curve of minimal degree. As an immediate application, we obtain irreducibility criteria for the space of minimal rational curves.

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